# Properties

 Label 1512.2.c.c Level 1512 Weight 2 Character orbit 1512.c Analytic conductor 12.073 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1512.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( -\beta_{2} + \beta_{3} ) q^{4} + ( \beta_{5} + \beta_{6} ) q^{5} - q^{7} + ( \beta_{4} + 2 \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( -\beta_{2} + \beta_{3} ) q^{4} + ( \beta_{5} + \beta_{6} ) q^{5} - q^{7} + ( \beta_{4} + 2 \beta_{6} ) q^{8} + ( 2 + \beta_{2} - \beta_{3} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{11} + ( -1 - \beta_{3} - 2 \beta_{7} ) q^{13} + \beta_{5} q^{14} + ( 3 - \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{17} + ( 1 + 4 \beta_{3} + 2 \beta_{7} ) q^{19} + ( -\beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{20} + ( -4 - \beta_{2} + 5 \beta_{3} - 2 \beta_{7} ) q^{22} + ( -\beta_{5} + \beta_{6} ) q^{23} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{25} + ( -4 \beta_{1} - \beta_{4} + \beta_{5} ) q^{26} + ( \beta_{2} - \beta_{3} ) q^{28} + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} -7 q^{31} + ( -2 \beta_{1} - 3 \beta_{5} ) q^{32} + ( -3 + \beta_{2} - 3 \beta_{3} - \beta_{7} ) q^{34} + ( -\beta_{5} - \beta_{6} ) q^{35} + ( 1 + 2 \beta_{3} + 2 \beta_{7} ) q^{37} + ( 4 \beta_{1} + 4 \beta_{4} - \beta_{5} ) q^{38} + ( -3 - 2 \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{40} + ( 4 \beta_{1} + 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{41} + ( 1 + 5 \beta_{3} + 2 \beta_{7} ) q^{43} + ( -4 \beta_{1} + 5 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{44} + ( 2 - \beta_{2} + \beta_{3} ) q^{46} + ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{47} + q^{49} + ( \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{50} + ( 1 + \beta_{2} + 7 \beta_{3} + \beta_{7} ) q^{52} + ( 6 + 2 \beta_{2} - \beta_{3} ) q^{55} + ( -\beta_{4} - 2 \beta_{6} ) q^{56} + ( 1 + \beta_{2} + \beta_{3} - \beta_{7} ) q^{58} + ( -3 \beta_{1} + 3 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{59} + ( 2 - 6 \beta_{3} + 4 \beta_{7} ) q^{61} + 7 \beta_{5} q^{62} + ( -3 \beta_{2} + 7 \beta_{3} ) q^{64} + ( 5 \beta_{1} + 5 \beta_{4} - \beta_{5} + \beta_{6} ) q^{65} + ( 3 + 3 \beta_{3} + 6 \beta_{7} ) q^{67} + ( -2 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{68} + ( -2 - \beta_{2} + \beta_{3} ) q^{70} + ( \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{71} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{73} + ( 4 \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{74} + ( -4 - \beta_{2} - 7 \beta_{3} - 4 \beta_{7} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{77} + ( 1 - 6 \beta_{2} + 3 \beta_{3} ) q^{79} + ( 2 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{80} + ( 2 - 3 \beta_{2} - 5 \beta_{3} - 4 \beta_{7} ) q^{82} + ( -3 \beta_{1} + 3 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{83} + ( 1 + 5 \beta_{3} + 2 \beta_{7} ) q^{85} + ( 4 \beta_{1} + 5 \beta_{4} - \beta_{5} ) q^{86} + ( -1 + 4 \beta_{2} + 4 \beta_{3} - 5 \beta_{7} ) q^{88} + ( 4 \beta_{1} + 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{89} + ( 1 + \beta_{3} + 2 \beta_{7} ) q^{91} + ( \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 3 - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{7} ) q^{94} + ( -8 \beta_{1} - 8 \beta_{4} + \beta_{5} - \beta_{6} ) q^{95} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{97} -\beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} + 16q^{10} + 28q^{16} - 24q^{22} + 8q^{25} - 56q^{31} - 20q^{34} - 28q^{40} + 16q^{46} + 8q^{49} + 4q^{52} + 48q^{55} + 12q^{58} - 16q^{70} - 24q^{73} - 16q^{76} + 8q^{79} + 32q^{82} + 12q^{88} + 36q^{94} - 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{3}$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$\nu^{4} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + 2 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 4$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 3 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{3} + 3 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$7 \beta_{6} + 6 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 0.178197 + 1.40294i 0.178197 − 1.40294i 1.40294 + 0.178197i 1.40294 − 0.178197i −1.40294 + 0.178197i −1.40294 − 0.178197i −0.178197 + 1.40294i −0.178197 − 1.40294i
−1.40294 0.178197i 0 1.93649 + 0.500000i 0.356394i 0 −1.00000 −2.62769 1.04655i 0 0.0635083 0.500000i
757.2 −1.40294 + 0.178197i 0 1.93649 0.500000i 0.356394i 0 −1.00000 −2.62769 + 1.04655i 0 0.0635083 + 0.500000i
757.3 −0.178197 1.40294i 0 −1.93649 + 0.500000i 2.80588i 0 −1.00000 1.04655 + 2.62769i 0 3.93649 0.500000i
757.4 −0.178197 + 1.40294i 0 −1.93649 0.500000i 2.80588i 0 −1.00000 1.04655 2.62769i 0 3.93649 + 0.500000i
757.5 0.178197 1.40294i 0 −1.93649 0.500000i 2.80588i 0 −1.00000 −1.04655 + 2.62769i 0 3.93649 + 0.500000i
757.6 0.178197 + 1.40294i 0 −1.93649 + 0.500000i 2.80588i 0 −1.00000 −1.04655 2.62769i 0 3.93649 0.500000i
757.7 1.40294 0.178197i 0 1.93649 0.500000i 0.356394i 0 −1.00000 2.62769 1.04655i 0 0.0635083 + 0.500000i
757.8 1.40294 + 0.178197i 0 1.93649 + 0.500000i 0.356394i 0 −1.00000 2.62769 + 1.04655i 0 0.0635083 0.500000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.c.c 8
3.b odd 2 1 inner 1512.2.c.c 8
4.b odd 2 1 6048.2.c.c 8
8.b even 2 1 inner 1512.2.c.c 8
8.d odd 2 1 6048.2.c.c 8
12.b even 2 1 6048.2.c.c 8
24.f even 2 1 6048.2.c.c 8
24.h odd 2 1 inner 1512.2.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.c 8 1.a even 1 1 trivial
1512.2.c.c 8 3.b odd 2 1 inner
1512.2.c.c 8 8.b even 2 1 inner
1512.2.c.c 8 24.h odd 2 1 inner
6048.2.c.c 8 4.b odd 2 1
6048.2.c.c 8 8.d odd 2 1
6048.2.c.c 8 12.b even 2 1
6048.2.c.c 8 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{4} + 8 T_{5}^{2} + 1$$ $$T_{17}^{2} - 10$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{4} + 16 T^{8}$$
$3$ 1
$5$ $$( 1 - 12 T^{2} + 71 T^{4} - 300 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$( 1 + 4 T^{2} + 111 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 378 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 24 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 14 T^{2} - 189 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 84 T^{2} + 2807 T^{4} + 44436 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 52 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 7 T + 31 T^{2} )^{8}$$
$37$ $$( 1 - 110 T^{2} + 5523 T^{4} - 150590 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 12 T^{2} + 2663 T^{4} + 20172 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 92 T^{2} + 4314 T^{4} - 170108 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 40 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 - 24 T^{2} + 3266 T^{4} - 83544 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 52 T^{2} - 522 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 20 T^{2} + 4218 T^{4} + 89780 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 236 T^{2} + 23871 T^{4} + 1189676 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 6 T + 140 T^{2} + 438 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 2 T + 24 T^{2} - 158 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 120 T^{2} + 13538 T^{4} - 826680 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 204 T^{2} + 25511 T^{4} + 1615884 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 4 T + 138 T^{2} + 388 T^{3} + 9409 T^{4} )^{4}$$